Geodesic
Inference about stationary distributions of Markov chains based on divergences with observed frequencies
Kybernetika, Tome 35 (1999) no. 3, pp. 265-280

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR   Zbl

For data generated by stationary Markov chains there are considered estimates of chain parameters minimizing $\phi $–divergences between theoretical and empirical distributions of states. Consistency and asymptotic normality are established and the asymptotic covariance matrices are evaluated. Testing of hypotheses about the stationary distributions based on $\phi $–divergences between the estimated and empirical distributions is considered as well. Asymptotic distributions of $\phi $–divergence test statistics are found, enabling to specify asymptotically $\alpha $-level tests.
For data generated by stationary Markov chains there are considered estimates of chain parameters minimizing $\phi $–divergences between theoretical and empirical distributions of states. Consistency and asymptotic normality are established and the asymptotic covariance matrices are evaluated. Testing of hypotheses about the stationary distributions based on $\phi $–divergences between the estimated and empirical distributions is considered as well. Asymptotic distributions of $\phi $–divergence test statistics are found, enabling to specify asymptotically $\alpha $-level tests.
Classification : 62E20, 62M02, 62M05
Keywords: $\phi$-divergence; empirical distributions; parameter estimation; hypotheses testing 
Menéndez, María Luisa; Morales, Domingo; Pardo, Leandro; Vajda, Igor. Inference about stationary distributions of Markov chains based on divergences with observed frequencies. Kybernetika, Tome 35 (1999) no. 3, pp. 265-280. http://geodesic.mathdoc.fr/item/KYB_1999_35_3_a0/
@article{KYB_1999_35_3_a0,
     author = {Men\'endez, Mar{\'\i}a Luisa and Morales, Domingo and Pardo, Leandro and Vajda, Igor},
     title = {Inference about stationary distributions of {Markov} chains based on divergences with observed frequencies},
     journal = {Kybernetika},
     pages = {265--280},
     year = {1999},
     volume = {35},
     number = {3},
     mrnumber = {1704667},
     zbl = {1274.62548},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_1999_35_3_a0/}
}
TY  - JOUR
AU  - Menéndez, María Luisa
AU  - Morales, Domingo
AU  - Pardo, Leandro
AU  - Vajda, Igor
TI  - Inference about stationary distributions of Markov chains based on divergences with observed frequencies
JO  - Kybernetika
PY  - 1999
SP  - 265
EP  - 280
VL  - 35
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/KYB_1999_35_3_a0/
LA  - en
ID  - KYB_1999_35_3_a0
ER  - 
%0 Journal Article
%A Menéndez, María Luisa
%A Morales, Domingo
%A Pardo, Leandro
%A Vajda, Igor
%T Inference about stationary distributions of Markov chains based on divergences with observed frequencies
%J Kybernetika
%D 1999
%P 265-280
%V 35
%N 3
%U http://geodesic.mathdoc.fr/item/KYB_1999_35_3_a0/
%G en
%F KYB_1999_35_3_a0

[1] Billingsley P.: Statistical methods in Markov chains. Ann. Math. Statist. 32 (1961), 12–40 | DOI | MR | Zbl

[2] Birch M. W.: A new proof of the Pearson–Fisher Theorem. Ann. Math. Statist. 35 (1964), 817–824 | DOI | MR | Zbl

[3] Bishop Y. M. M., Fienberg S. E., Holland P. W.: Discrete Multivariate Analysis. Theory and Practice. The MIT Press, Cambridge, Massachusetts 1975 | MR | Zbl

[4] Cressie N., Read T. R. C.: Multinomial goodness of fit tests. J. Royal Statist. Soc., Ser. B 46 (1984), 440–464 | MR | Zbl

[5] Drost F. C., Kallenberg W. C. M., Moore D. S., Oosterhoff J.: Power approximations to multinomial tests of fit. J. Amer. Statist. Assoc. 84 (1989), 130–141 | DOI | MR | Zbl

[6] Glesser L. J., Moore D. S.: The effect of dependence on chi–squared and empiric distribution tests of fit. Ann. Statist. 11 (1983), 1100-1108 | MR

[7] Glesser L. J., Moore D. S.: The effect of positive dependence on chi–squared tests for categorical data. J. Royal Statis. Soc., Ser. B 47 (1983), 459–465 | MR

[8] Liese F., Vajda I.: Convex Statistical Distances. Teubner, Leipzig 1987 | MR | Zbl

[9] Menéndez M. L., Morales D., Pardo L., Vajda I.: Divergence–based estimation and testing of statistical models of classification. J. Multivariate Anal. 54 (1996), 329–354 | DOI | MR

[10] Menéndez M. L., Morales D., Pardo L., Vajda I.: Testing in stationary models based on $f$–divergences of observed and theoretical frequencies. Kybernetika 33 (1997), 465–475 | MR

[11] Moore D. S.: The effect of dependence on chi–squared tests of fit. Ann. Statist. 10 (1982), 1163–1171 | DOI | MR | Zbl

[12] Morales D., Pardo L., Vajda I.: Asymptotic divergence of estimates of discrete distributions. J. Statist. Plann. Inference 48 (1995), 347–369 | DOI | MR | Zbl

[13] Read T. R. C., Cressie N. A. C.: Goodness–of–Fit Statistics for Discrete Multivariate Data. Springer, Berlin 1988 | MR | Zbl

[14] Salicrú M., Morales D., Menéndez M. L., Pardo L.: On the applications of divergence type measures in testing statistical hypotheses. J. Multivariate Anal. 51 (1994), 372–391 | DOI | Zbl

[15] Tavaré S., Altham P. M. E.: Serial dependence of observations leading to contingency tables, and corrections to chi–squared statistics. Biometrika 70 (1983), 139–144 | DOI | MR | Zbl